15 UCF Eample 1.11 Assume that vector field: A0 A= a r r eists in a region surrounding the origin of a spherical coordinate sstem. Find the value of the closed-surface integral over the unit sphere.

16 UCF Eample 1.11 Solution: ds r sinθdθdφa = r The closed-surface integral is given b A = r sin d d = 4πA φ= 0 θ= 0 r φ= π θ= π 0 A ds a ( θ θ φ ) r a r rdθ In this integral, we have used the differential surface area in spherical coordinates that has a unit vector a r. If the vector A had an additional components directed in the a θ or a φ directions, their contribution to this surface integral would be zero, since the scalar product of these terms will be equal to zero. 0 e111.m rsinθdφ

17 UCF Eample 1.1 A= a + a + za 5 4 Assume that the vector field z is defined in a region 1 3, 4 and 1 z Find the value of the closed-surface integral A ds over the surface of this region. e11.m

19 UCF Eample 1.1 On the right surface, and is the onl component perpendicular to that surface. Therefore, A ds = A ddz right = 4 = ddz = ddz A ds right = 16ddz = 16 dz = On the left surface, and is the onl component perpendicular to that surface. Therefore, A ds = A ( ddz) left = ( ) ( ddz) 4ddz = = A ds left = 4ddz = 4 dz = z z A= a + a + a 1 3, 4 and 1 z

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